presentation - Dept. of IE, CUHK Personal User Web Server

Concatenated Polar CodesMayank BakshiCaltechSidharth JaggiCUHKMichelle EffrosCaltech

Typical multiuser systemChannelPolar Codes:[Arikan’09, Korada et al’09, Hussami et al ’09]Capacity achieving:Many point-to-point channelsMulti-user channels- MAC, degraded broadcast, Gelfand-Pinsker- Capacity bounds : known in many cases- Practical coding schemes not known formostchannelsKey challenges:Network Source Coding- Slepian-Wolf, coded side information...- encoding/decoding complexity- good error probability decay

Typical multiuser systemChannelPolar Codes:[Arikan’09, Korada et al’09, Hussami et al ’09]Capacity achieving:Many point-to-point channelsMulti-user channels- MAC, degraded broadcast, Gelfand-Pinsker- Capacity bounds : known in many cases- Practical coding schemes not known formostchannelsKey challenges:- encoding/decoding complexityNetwork Source Coding- Slepian-Wolf, coded side information...Encoding complexity: O(n log n)Decoding complexity: O(n log n)Error probability: 2 −√ n- good error probability decay

Point-to-point channel: other codesEncodingcomplexityDecodingcomplexityP eO(N log N) O(N log N)Random Codesexp(−θ(N))exp(O(N))exp(O(N))Spielman Codesexp(−θ(N))θ(N)θ(N · 2 1/ )LP DecodingPolar Codesexp(−θ(N))exp(−o( √ N))O(N log N) O(N log N)Modified PolarCodesexp(−o(N β(l) )) O(lNlog N) O(2 l N log N)Desirableexp(−θ(N))O(N log N) O(N log N)

Improve error probability?Desirable exp(−θ(N)) O(N log N) O(N log N)Polar Codes exp(−o( √ N)) O(N log N) O(N log N)

Improve error probability?Desirable exp(−θ(N)) O(N log N) O(N log N)Polar Codes exp(−o( √ N)) O(N log N) O(N log N)Concatenation [Forney ’66]u 1u 2u kPx 1x 2y 1y 2P−1u 1u 2R-SPP −1−1R-SPP −1u kx ny nEncodingDecoding

EncodingmR OnR IN = m · n...overallblocklength

EncodingmR OnR IN = m · n...overallblocklengthR = R O · R IK =(mR O ) · (nR I )messageblocklengthoutercodeinnercode

...EncodingN = m · noverallblocklengthmR OnR InR IR-S encoderR = R O · R ImK =(mR O ) · (nR I )......messageblocklengthoutercodeinnercode

...EncodingN = m · noverallblocklengthmR OnR InR IR-S encoderR = R O · R ImK =(mR O ) · (nR I )......messageblocklengthoutercodeinnercodePolar encoderm......n

DecodingmR OnR InR I...mR-S decoder......mPolar decoder......n

Does it work well?mR OnR InR I...mR-S encoder......mPolar encoder......n

Does it work well?mR OnR InR I...First attempt:m = N/log Nn = log NmR-S encoder......mPolar encoder......n

Does it work well?mR OnR InR I...First attempt:m = N/log Nn = log NR-S encodermError Probability:......Error if more thanm(1 − 2R O )inner codes failPolar encoderm......n

Does it work well?mR OnR InR I...First attempt:m = N/log Nn = log NR-S encodermError Probability:......Error if more thanPr(error) ∼ exp(−O(m(1 − 2R O )Nlog N ))inner codes failmPolar encoder......n

Does it work well?mR OnR InR I...First attempt:m = N/log Nn = log NR-S encodermError Probability:......Error if more thanPr(error) ∼ exp(−O(m(1 − 2R O )Nlog N ))inner codes failmPolar encoderEncoding complexity: O(N log N)......n

Does it work well?mR OnR InR I...First attempt:m = N/log Nn = log NR-S encodermError Probability:......Error if more thanPr(error) ∼ exp(−O(m(1 − 2R O )Nlog N ))inner codes failmPolar encoderEncoding complexity: O(N log N)Decoding complexity: O(N 2 log N)......n

Can decoding be performed using encoding?mR OnR InR I...mR-S encoder......mPolar encoder......n

Can decoding be performed using encoding?mR OnR InR I...mSystematic R-S encoder......mPolar encoder......n

mDecoding... ...n

mDecoding......nPolar decoderm......nR I

mDecoding......nPolar decodermR O m(1 − R O )......nR I

mDecoding... ...nPolar decodermR O m(1 − R O )... ...nR ISystematic R-S encodermR Om(1 − R O )......nR I

mDecoding... ...nPolar decodermR O m(1 − R O )... ...nR ImR OSystematic R-S encoder ...= ?...m(1 − R O )......nR I

mDecoding... ...nPolar decodermR O m(1 − R O )... ...nR ISystematic R-S encoder... ...= ? ...mR O...m(1 − R O )...nR IYesmR OnR I

mDecoding... ...nPolar decodermR OmR O m(1 − R O )R-S decoder...nR I... ...nR INoSystematic R-S encoder... ...= ? ...mR Om(1 − R O )mR OYes......nR InR I

mDecoding... ...nDecoding complexity:P s x encoding complexity+ x R-S decoder complexity(1 − P s )Polar decodermR OmR O m(1 − R O )... ...nR IR-S decoder...nR I1 − P sNoSystematic R-S encoder... ...= ? ...mR Om(1 − R O )mR OP sYes......nR InR I

Does it help?mR OnR InR I...First attempt:m = N/log Nn = log NSystematic R-S encodermError Probability:......Error if more thanPr(error) ∼ exp(−O(m(1 − 2R O )Nlog N ))inner codes failmPolar encoderEncoding complexity: O(N log N)......n

Does it help?......mR OnR InR IFirst attempt:m = N/log Nn = log NSystematic R-S encodermError Probability:...Error if more thanPr(error) ∼ exp(−O(m(1 − 2R O )Nlog N ))inner codes failPolar encoderEncoding complexity: O(N log N)mDecoding complexity:......nP s x encoding complexity+ (1 − P s ) x R-S decoder complexity

Does it help?......mR OnR InR IFirst attempt:m = N/log Nn = log NSystematic R-S encodermError Probability:...Error if more thanPr(error) ∼ exp(−O(m(1 − 2R O )Nlog N ))inner codes failPolar encoderEncoding complexity: O(N log N)mDecoding complexity:......nP s x encoding complexity+ (1 − P s ) x R-S decoder complexity= O(N 2 log N)

Can it ever help?mR OnR InR I...smaller m?mSystematic R-S encoder......mPolar encoder......n

Can it ever help?mR OnR InR I...smaller m?m = N α (α < 1)n = N 1−αmSystematic R-S encoder......mPolar encoder......n

Can it ever help?mR OnR InR I...smaller m?m = N α (α < 1)n = N 1−αmSystematic R-S encoder......Polar encodermDecoding complexity:......nP s x encoding complexity+ (1 − P s ) x R-S decoder complexity= O(N log N)

Can it ever help?mR OnR InR I...smaller m?m = N α (α < 1)n = N 1−αmSystematic R-S encoder......Polar encoderEncoding complexity: O(N log N)mDecoding complexity:......nP s x encoding complexity+ (1 − P s ) x R-S decoder complexity= O(N log N)

Can it ever help?......mR OnR InR Ismaller m?m = N α (α < 1)n = N 1−αSystematic R-S encodermError Probability:...Error if more thanm(1 − 2R O )Pr(error) ∼ exp(−O(N α/2 ))inner codes failPolar encoderEncoding complexity: O(N log N)mDecoding complexity:......nP s x encoding complexity+ (1 − P s ) x R-S decoder complexity= O(N log N)

desirable: exp(−θ(N)) O(N log N) O(N log N)mnP eEncodingcomplexityDecodingcomplexityθ(N α ) θ(N 1−α )exp(−o(N α/2 )) O(N log N)O(N log N)θ(Nlog N )θ(log N)exp(−O(Nlog N ))O(N log N) O(N 2 log N)

desirable: exp(−θ(N)) O(N log N) O(N log N)mnP eEncodingcomplexityDecodingcomplexityθ(1)θ(N)θ(N α ) θ(N 1−α ) exp(−o(N α/2 )) O(N log N)O(N log N)θ(θ(θ(θ(Nlog 4 N )Nlog 3 N )θ(log 4 N)Nlog 2 N ) θ(log 2 N)θ(log 3 N)Nlog N ) θ(log N)exp(−O(Nlog N ))O(N log N) O(N 2 log N)

desirable: exp(−θ(N)) O(N log N) O(N log N)mnP eEncodingcomplexityDecodingcomplexityθ(1) θ(N) exp(−o( √ N) O(N log N) O(N log N)θ(N α ) θ(N 1−α ) exp(−o(N α/2 )) O(N log N)O(N log N)Nθ( θ(log 4 N)O(N log N) O(N log N)log 4 N )Nexp(−O(log 4 N ))Nθ(log 3 N ) Nexp(−Ω(log 27/8 N )) O(N log N) O(N log N)Nθ(exp(−O(O(N log N)log 2 N ))O(N 2 log N)θ(Nlog 2 N ) θ(log 2 N)θ(log 3 N)Nlog N ) θ(log N)exp(−O(Nlog N ))O(N log N) O(N 2 log N)

desirable: exp(−θ(N)) O(N log N) O(N log N)mnP eEncodingcomplexityDecodingcomplexityθ(1)θ(N)exp(−o( √ N) O(N log N) O(N log N)θ(N α ) θ(N 1−α )exp(−o(N α/2 )) O(N log N)O(N log N)θ(θ(θ(θ(Nlog 4 N )Nlog 3 N )θ(log 4 N)Nlog 2 N ) θ(log 2 N)θ(log 3 N)Nlog N ) θ(log N)Nexp(−O(log 4 N ))Nexp(−Ω(log 27/8 N ))exp(−O(exp(−O(Nlog 2 N ))Nlog N ))O(N log N) O(N log N)O(N log N) O(N log N)O(N log N) O(N 2 log N)O(N log N) O(N 2 log N)

Concatenated polar codesAchieve capacity for arbitrary point-to-point channelsEncoding Complexity:O(N log N)...Decoding Complexity:Error probability:O(N log N)−N/(log N)3.3752Systematic R-S encoder......Polar encoder......

Concatenation in multi-user channelse.g. Multiple access channelXZp(y|x, z)Y

Concatenation in multi-user channelse.g. Multiple access channelXZp(y|x, z)Y...Systematic R-S encoder- Apply concatenation to each message- Polar inner code......Polar encoder......

Concatenation in multi-user channelse.g. Multiple access channelXZp(y|x, z)Y...Systematic R-S encoder- Apply concatenation to each message- Polar inner code......Achieve capacityEncoding Complexity:Decoding Complexity:O(N log N)O(N log N)Polar encoderError probability:−N/(log N)3.3752......

Concatenation in network source codinge.g. Coded side informationXXY(X, Y) ∼ p(x, y)

Concatenation in network source codinge.g. Coded side information.... ..X...XSystematic R-S encoder.... ..Y(X, Y) ∼ p(x, y)... ...

Concatenation in network source codinge.g. Coded side information.... ..X...XSystematic R-S encoder.... ..Y(X, Y) ∼ p(x, y)... ...- Systematic bits ∼ p(x, y)

Concatenation in network source codinge.g. Coded side information.... ..X...XSystematic R-S encoder.... ..Y(X, Y) ∼ p(x, y)... ...- Systematic bits ∼ p(x, y)- Parity bits: not even i.i.d.- Polar code may not work

Concatenation in network source codinge.g. Coded side information.... ..X...XSystematic R-S encoder.... ..Y(X, Y) ∼ p(x, y)......- Systematic bits ∼ p(x, y)- Parity bits: not even i.i.d.- Polar code may not work- Transmit parity bits without coding!...Polar encoder...

Concatenation in network source codinge.g. Coded side information..........X...XSystematic R-S encoder..........Y......(X, Y) ∼ p(x, y)Polar encoderAchieve optimal ratesEncoding Complexity:Decoding Complexity:O(N log N)O(N log N)......Error probability:−N/(log N)3.3752

Concluding remarksKey ideas• Concatenate: Systematic R-S outer, Polar inner• Suitable inner/outer codelength => reduced average decoding complexity• Modification for source coding: apply source code only to message bits, not parity

Concluding remarksKey ideas• Concatenate: Systematic R-S outer, Polar inner• Suitable inner/outer codelength => reduced average decoding complexity• Modification for source coding: apply source code only to message bits, not parityResults• Efficient, capacity achieving codes• Arbitrary point-to-point• Several multi-user channels: Degraded broadcast channel, multiple-accesschannel• Network Source coding problems: e.g. Slepian-Wolf, Coded Side Information

Concluding remarksDesirable exp(−θ(N)) O(N log N) O(N log N)Polar Codes exp(−o( √ N)) O(N log N) O(N log N)ConcatenatedCodesexp(−Ω(Nlog 27/8 N )) O(N log N) O(N log N)Next Steps• Use insights from concatenated code to design a better single stage code• Joint decoding improves the performance in some cases• Other parameters?•Dependence of error probability on coding rate ?